How to make this animation look more like a satellite orbiting a binary star?
There are three things I'd recommend.

One is making your background black. Any other color will artificially reduce the contrast, making the stars (and Suzanne) seem less bright than they are.

Another is improving Suzanne's material. It should be a correct, Fresnel diffuse (if diffuse is what you're going for), and it should be a lighter color, something closer to white. It looks dark green at the moment. Dark colors don't reflect as much light as light colors (that's why they're dark ;-) ), so the light source has to be brighter. Colored materials react to colored lights in funny ways. In particular, your green material is going to be preferential to the white light, over the red light, making the red light look dimmer, even if it's the same intensity.

For a (mostly) correct Fresnel diffuse shader, just mix a diffuse shader with a glossy shader, plugging a fresnel input node into the factor slot on the mix shader. Fresnel in factor, diffuse in top slot, glossy in bottom slot, mix piped to surface. This will make her reflect light a bit more reasonably.

Lastly, if you need a sun lamp to "see what you're doing" then something is wrong. It's necessarily going to mask/diminish/hide the effect of the two stars. It won't LOOK like Suzanne is being lit by the pulsar because her dark side isn't DARK... it's being lit by the sun lamp. If you want to make the depth more obvious, decrease your focal length, and re-position the camera.

Hope that helps!